Mobile plate accelerometer with electrostatic feedback motor

ABSTRACT

A slave mobile plate accelerometer using variations of capacitance to detect the movement of a mass. The apparatus comprises at least one pair of fixed electrodes rigidly attached to an armature, and at least one mobile electrode suspended by springs from the armature, between the two fixed electrodes of each pair of fixed electrodes, to form two capacitors, each mobile electrode being adapted to move between the fixed electrodes of each pair of fixed electrodes, due to the effect of acceleration, so causing a variation of the capacitance of each capacitor. The accelerometer further includes an electronic circuit for adjusting the electrostatic stiffness of at least one combination of a fixed electrode and a mobile electrode and a control system for detecting the variation of the capacitance of each capacitor.

The invention relates to the field of acceleration sensors. Theinvention is more particularly a slaved mobile plate accelerometer usingcapacitance variations to detect the movement of a mass.

Micromachined accelerometers are now well known in the sensor industry.They generally comprise a mobile plate, referred to as the seismicplate, suspended from a fixed armature by springs; the mobileplate/armature/spring assembly is obtained by chemically etching asilicon wafer.

The mobile plate comprises an electrode, for example. The fixed armaturecomprises two electrodes. The mobile electrode forms a capacitor witheach of the two fixed electrodes. If acceleration is applied to thesensor, the mobile plate moves relative to the fixed armature, socreating an imbalance of the capacitances. A control circuit detects thedifference between the two capacitances and reacts by applying afeedback voltage, and therefore an electrostatic force, between themobile plate electrode and the fixed armature electrodes, in order toreturn the mobile plate to its original position. This provides anelectrostatic feedback motor.

The electrostatic stiffness is known to be a function of theelectrostatic force. For a sensor of the above kind, the principle ofadjusting the electrostatic stiffness is as follows.

In the absence of electric field, a force F_(m) tends to return themobile electrode to its rest position by virtue of the stiffness k_(m)of the springs (k_(m) is the derivative of the return force with respectto the displacement). However, in the presence of an electric field(voltage between the fixed electrodes and the mobile electrode), anelectrostatic force F_(e) attracts the mobile electrode towards one orother of the fixed electrodes. This force is the sum of two forces inopposite directions proportional to the square of the electric fieldbetween the electrodes:

F_(e)=ε₀·(S₁·E₁ ²−S₂·E₂ ²), where S_(i) is the surface area of thecapacitor formed by the fixed electrode i and the mobile electrode,E=V_(i)/d_(i), V_(i) is the voltage between the fixed plate electrode iand the electrode of the facing mobile plate, and d_(i) is the distancebetween the electrode i and the mobile plate electrode (i=1 or 2).

The electrostatic force is a function of the distance between theelectrodes, and therefore of the displacement, like the spring returnforce, but in the opposite direction. Its derivative with respect to thedisplacement is the electrostatic stiffness k_(e).

It is also known that the frequency f_(rm) of mechanical resonance isrelated to the mass m of the mobile plate and to the stiffness k_(m) ofthe springs: $f_{rm} = {\frac{1}{2\pi}\sqrt{\frac{k_{m}}{m}}}$

The mass of the mobile plate is generally known accurately and closelycontrolled in manufacture, but this is not the case with the stiffnessk_(m) of the springs. Performance therefore varies greatly from onemanufacturing batch to another. This makes it necessary to solve adifficult problem in the manufacture of such accelerometers, namely thatof obtaining a precise and reproducible frequency of mechanicalresonance of the mobile plate/spring cell. The dynamic range (maximumsignal-to-noise ratio) of this type of sensor is highly dependent onthis frequency.

There is also another problem to be solved for accelerometers sensitiveto the vertical component: sagging of the mobile mass due to the effectof gravity.

In vertical accelerometers (i.e. ones sensitive to the verticalcomponent of acceleration), the mass sags due to its own weight by anamount Δz=m·g/k_(m) (m=mass of mobile plate, g=acceleration due togravity, k_(m)=mechanical stiffness of springs).

Moreover, as the dynamic range of the accelerometer is proportional tothe mass divided by the stiffness (S=m/k), a high mass and a lowstiffness are required to achieve good performance. This leads to alarge sag due to gravity.

In the vertical position, for a system with free deformation, thedistance between the fixed plates and the mobile plates must be at leastequal to the sag, as otherwise return systems must be used. However, toogreat a distance between the electrodes leads to poor performancebecause of problems with obtaining sufficient electric fields. Thesagging therefore limits the performance of the sensor.

The mobile mass is generally centred on the armature in an attempt tosolve these problems, which imposes the use of relatively complexmanufacturing techniques such as:

1. prestressing the springs,

2. additional compensator springs,

3. electrostatic return (see U.S. Pat. No. 5,345,824),

4. remote electromagnetic return, and

5. additional fabrication steps (see U.S. Pat. No. 4,922,756).

U.S. Pat. No. 4,922,756 describes the fabrication of a micromachinedsensor with the stiffness constants of the springs precisely controlledduring the fabrication process. However, this is achieved at the priceof additional fabrication technology steps.

U.S. Pat. No. 5,345,824 addresses the problem of the spread of thestiffness constant of the springs by minimizing the mechanical stiffnessconstant k_(m) but centring the mass by means of a small percentage ofthe total available electrostatic force. The output signal is thereforeindependent of the constant k_(m) because the springs are not deflected.However, this modifies the sensitivity of the accelerometer.

The skilled person therefore usually seeks a compromise betweenoptimizing performance by reducing the stiffness constant, whichincreases the sag, optimizing performance by reducing the distancebetween the electrodes to obtain sufficiently high electric fields,which limits the possibilities of sagging, and optimizing the usablerange of frequencies.

The device according to the invention is a slaved mobile plateaccelerometer using variations of capacitance to detect the movement ofa mass, it comprises:

at least one fixed electrode rigidly attached to an armature,

at least one mobile electrode suspended by springs from the armature andfacing each fixed electrode to form at least one capacitor, each mobileelectrode being adapted to move relative to each fixed electrode due tothe effect of acceleration, so causing a variation of the capacitance ofeach capacitor, and

an electronic circuit for adjusting the electrostatic stiffness andcomprising a control system for detecting the variation of thecapacitance of each capacitor and reacting by applying a feedbackvoltage between each mobile electrode and the fixed electrode facingsaid mobile electrode,

characterized in that the springs have a stiffness chosen intentionallyto place the mechanical resonant frequency beyond the top frequency ofthe band of interest and the circuit for adjusting the electrostaticstiffness is adapted to return the apparent resonant frequency into theband of interest.

A stiffness-adjusting device according to the invention of the abovekind improves the performance of the system since it simultaneously:

1. compensates the spread of the mechanical stiffness of the springssuspending the mobile plate,

2. limits the sagging of vertical accelerometers by using a highmechanical stiffness, compensated by a high electrostatic stiffness, and

3. optimizes performance in the wanted band.

Note that deliberately placing the frequency of mechanical resonancebeyond the top frequency of the wanted band overcomes the prejudices ofthe skilled person, this solution being at first sight unfavourable tothe dynamic range, which is proportional to S=m/k_(m).

In an advantageous embodiment of the accelerometer according to theinvention, the accelerometer includes two fixed electrodes electricallyinsulated from each other.

In another advantageous embodiment of the accelerometer according to theinvention the accelerometer includes a single mobile electrode.

In another advantageous embodiment, the electronic circuit of theaccelerometer according to the invention enables time-divisionmultiplexing of each mobile electrode.

In a further advantageous embodiment, the accelerometer according to theinvention has a time-division multiplexing cycle which includes foursteps:

a first step during which a voltage sample and its symmetricalcounterpart relative to ground are respectively applied between eachfixed electrode and the mobile electrode,

a second step during which the capacitor constituted by one of said twofixed electrodes and the mobile electrode and the capacitor constitutedby the other fixed electrode and the mobile electrode are discharged,

a third step during which a feedback voltage is applied to one or theother of the capacitors constituted by the mobile electrode and one ofthe fixed electrodes, as a function of a decision taken by the controlsystem, and

a fourth step during which the operation of the second step is repeated.

In another advantageous embodiment, the electronic circuit of theaccelerometer according to the invention varies the amplitudes of thevoltages between each fixed electrode and each mobile electrode toadjust the electrostatic stiffness.

A further advantageous embodiment of the electronic circuit of theaccelerometer according to the invention enables the duration of thetime-division multiplexing steps to be adjusted.

In a further advantageous embodiment of the accelerometer according tothe invention, the duration of the time-division multiplexing steps isadjusted to adjust the electrostatic stiffness.

In a further advantageous embodiment of the accelerometer according tothe invention, the electrostatic stiffness is adjusted to compensate thespread of the mechanical stiffness of the springs.

In another advantageous embodiment of the accelerometer according to theinvention, the electrostatic stiffness is adjusted to reduce saggingwhen the accelerometer is in a vertical position, without loss ofperformance.

In another embodiment of the accelerometer according to the invention,the electrostatic stiffness is adjusted to optimize performance as afunction of the wanted band.

The accelerometer according to the invention can also be a component ofanother, more complex device.

The following description of one particular embodiment of the inventionis purely illustrative and is not limiting on the invention. It must beread with reference to the accompanying drawings:

FIG. 1 is a diagram showing a particular non-limiting embodiment of asensor of an accelerometer according to the invention.

FIG. 2 is a diagrammatic representation of a capacitive device of thesensor.

FIG. 3 is a block diagram of one particular but non-limiting embodimentof an accelerometer according to the invention.

FIG. 4 is a schematic representation of one example of a mobileelectrode multiplexing period.

The invention, which can be constructed and operate in accordance withthe following non-limiting specific embodiment, comprises a sensor 1 andan electronic circuit 10.

The sensor 1 comprises the following components:

an armature 2,

two fixed electrodes 3 and 4 rigidly attached to the armature 2,

a mobile electrode 5 supported by a mobile plate 6, and

springs 7 elastically connecting the mobile plate 6 to the armature 2.

The mobile plate 6 which supports the mobile electrode 5, the armature 2to which the fixed electrodes 3 and 4 are connected and the springs 7are micromachined in a semiconductor substrate by the usualmicrofabrication technology processes of microelectronics: chemicaletching, ion erosion, photolithography or electrolithography, ionimplantation, etc. The semiconductor can be silicon, for example.

The mobile electrode 5 is formed by a single conductive area or aplurality of conductive areas which are electrically interconnected.

Each of the fixed electrodes 3 and 4 is formed by a single conductivearea or a plurality of conductive areas which are electricallyinterconnected. The fixed electrodes 3 and 4 as described are thereforeelectrically insulated from each other.

The mobile electrode 5 is electrically insulated from the fixedelectrodes 3 and 4.

The mobile electrode 5 forms a capacitor with each of the fixedelectrodes 3 and 4. The sensor 1 therefore comprises two capacitors 8and 9 having the mobile electrode 5 as a common electrode.

The mobile electrode 5 must move between the two fixed electrodes 3 and4 due to the effect of a force. Accordingly, if an acceleration isapplied to the sensor 1, the displacement of the mobile electrode 5simultaneously varies the capacitances of the capacitors 8 and 9.

In an advantageous but non-limiting embodiment of the invention, theelectronic circuit 10 is of the “switched capacitor” type. It alsocomprises voltage generators 14, 15 and 19, a capacitor 21, an amplifier22 and a control system 20. It is timed by a clock 30.

Four switches 11, 12, 13 and 17 selectively switch certain components ofthe electronic circuit 10 into and out of circuit during certain phasesof the cycle of the clock 30.

The switch 11 selectively connects the fixed electrode 3 to ground, to+V_(m) 14 or to a circuit 16 generating a feedback voltage V_(cr). Theswitch 12 connects the fixed electrode 4 selectively to ground, to−V_(m) 15 or to the circuit 16 generating the feedback voltage V_(cr).The switch 13 selectively connects the mobile electrode 5 to the inputof the control system 20 or to ground. The switch 19 connects thegenerator 19 of the voltage V_(ref) into circuit.

The switches 11, 12 and 13 are also used to ground the electrodes 3, 4and 5 to discharge the capacitors 8 and 9, respectively. The capacitor21 is also discharged by a switch (not shown).

A parallel RC circuit 18 is advantageously incorporated into the circuit16 generating the feedback voltage V_(cr). The parallel RC circuit 18comprises a discharge capacitor C_(d) 23 and a discharge resistor R_(d)24.

The parallel RC circuit 18 shunts the generator 19 of the voltageV_(ref). The switch 17 connects the parallel RC circuit 18 to thegenerator 19 of the voltage V_(ref).

As explained below, the control system 20 detects the variation of thecapacitance of the capacitors 8 and 9 by applying voltages +V_(m) and−V_(m) between the mobile electrode 5 and the fixed electrodes 3 and 4and by applying a feedback voltage +V_(cr) between the mobile electrode5 and one of the fixed electrodes 3 or 4, the mobile electrode/fixedelectrodes assembly then constituting an electrostatic feedback motorwhich returns the mobile electrode 5 to its initial position. Thecontrol system 20 can include a Sigma-Delta modulator.

The cycle of the clock 30 controls multiplexing of the mobile electrode5. FIG. 4 shows one multiplexing period. That period defines a timeinterval P.

At time T₁ of the multiplexing period, the switches 11 and 12 connectthe electrodes 3 and 4 to the voltage generators 14 and 15. The switch13 connects the mobile electrode 5 to the capacitor 21. This position ofthe switches 11, 12 and 13 is maintained until time T₂. The timeinterval between T₁ and T₂ constitutes step 1: detection of the positionof the mobile electrode 5.

The switches 11, 12 and 13 respectively connect the fixed electrodes 3and 4 and the mobile electrode 5 to ground at time T₂ and remain in thisconfiguration until time T₃. The time interval between T₂ and T₃ definesstep 2.

The switches 11 and 12 connect the electrodes 3 or 4 to the voltagegenerator 16 at time T₃ and remain in this configuration until time T₄.The time interval between T₃ and T₄ defines step 3: applying thefeedback force which has a motor effect on the mobile electrode 5.

The switches 11 and 12 connect the fixed electrodes 3 and 4 to ground attime T₄ and remain in this configuration until time T₁+P. The timeinterval between T₄ and T₁+P defines step 4.

The cycle of the clock 30 therefore defines four steps:

Step 1: The position of the mobile electrode 5 is measured by applyingtwo voltages +V_(m) and −V_(m) symmetrical about ground between themobile electrode 5 and the fixed electrodes 3 and 4, respectively. Ifthe mobile electrode 5 is not centred, the capacitance C₁ of thecapacitor 8 and the capacitance C₂ of the capacitor 9 are not equal, andthis results in a transfer of charge ΔQ=V_(m) (C₁−C₂) into the capacitor21. The voltage V_(s) represents the position of the mobile electrode 5at the end of step 1.

Step 2: The capacitors 8 and 9 are discharged.

Step 3: A feedback voltage V_(cr) is applied to one of the capacitors 8or 9, according to a decision taken by the control system 20, whichdecision is itself a function of the difference between the position ofthe mobile electrode 5 and its nominal position (that in which C₁=C₂) .This voltage V_(cr) develops an electrostatic force F_(e) which tends toreturn the mobile electrode 5 to its nominal position.

Step 4: The capacitors 8 and 9 are discharged.

Discharging the capacitors 8 and 9 eliminates interference between themeasurement of the position and the application of the feedback voltage.

The duration of each these steps can be controlled.

The feedback voltage V_(cr) can be constant, i.e. applied to theelectrodes 3 or 4 and 5 in the form of a square pulse, but it is moreadvantageous if the feedback voltage V_(cr) is applied to the electrodes3 or 4 and 5 in the form of a pulse with a steep rising edge and anexponentially decreasing falling edge.

This reduces the sensitivity to clock fronts, i.e. to clock phase noise,and therefore improves the signal/noise ratio of the sensor. Theelectrostatic force F_(e) applied to the mobile electrode 5 isproportional to the integral of The square of the feedback voltageV_(cr) . Phase noise on the falling edge of the pulse thereof does notgreatly change the value of this integral if the feedback voltage V_(cr)falls progressively, possibly even to zero, at the end of the pulse.Thus a decrease of the exponential type or like that of a damped sinefunction is also suitable, for example.

In this case, the feedback voltage V_(cr) is advantageously generated inthe following manner: The capacitor C_(d) 23 is charged to a referencevoltage V_(ref) by a generator 19 of the voltage V_(ref) during steps 1,2 and/or 4. During step 3, which is the step of application of thefeedback voltage V_(cr), the capacitor C_(d) 23 is connected between themobile electrode 5 and one or other of the fixed electrodes 3 and 4 andto the terminals of the discharge resistor R_(d) 24. The decision toapply the feedback voltage V_(cr) to one or other of the fixedelectrodes 3 and 4 is taken according to the sign of the displacement ofthe mobile plate 6.

The principle of adjusting the resonant frequency of the accelerometeraccording to the invention is described below.

The apparent resonant frequency f_(ra) is related to the mass of themobile plate and to the apparent return stiffness:$f_{ra} = {{\frac{1}{2\pi}\sqrt{\frac{k_{a}}{m}}{where}\quad k_{a}\quad \left( {{apparent}\quad {stiffness}} \right)} = k_{m}}$

(spring stiffness−k_(e) (electrostatic stiffness) and where m is themass of the mobile plate.

The value of k_(a) can therefore be adjusted by varying theelectrostatic stiffness k_(e).

The performance of the sensor is improved because it is related to theapparent frequency f_(ra) and not to the mechanical resonant frequency$f_{rm} = {\frac{1}{2\pi}\sqrt{\frac{k_{m}}{m}}}$

and because k_(a) can be reduced, i.e. it is possible to increase thedynamic range, which is proportional to S=m/k_(a).

During the production of the accelerometer according to the invention,the mechanical resonant frequency is deliberately placed beyond the topfrequency of the band of interest (high stiffness k_(m)). This limitssagging and reduces the distance between electrodes, and thereforeenables the use of high electric fields (and therefore a highelectrostatic stiffness k_(e)) to bring the apparent resonant frequencyback into the band of interest.

The feasible multiplexing frequency range is from 100 to 500 times thehighest wanted frequency, for example.

This facilitates obtaining a high loop gain in the band of interestwithout compromising the stability of the looped system.

In one particular but non-limiting method of adjusting the accelerometeraccording to the invention, the electrostatic stiffness k_(e) isprovided by adjusting the duration of step 1. It is preferable to adjustthe duration of step 1 because this affects only the parameter k_(e).

In continuous systems the electrostatic stiffness, i.e. the derivativeof the electrostatic force with respect to the distance between theelectrodes, is proportional to the square of the voltage and inverselyproportional to the cube of the distance between the electrodes. Insampled systems, because the sampling frequency is very much higher thanthe cut-off frequency, it is also proportional to the cyclic ratio ofapplication of the voltage (Ta/Te where Ta is the period of applicationof the feedback voltage V_(cr) and Te is the sampling period).

In other embodiments of the present invention the amplitudes of thevoltages between the fixed electrodes 3 and 4 and the mobile electrode 5could be varied by the electronic circuit 10. This can also adjust theelectrostatic stiffness, even if in this case the performance of theaccelerometer is less good than when the electrostatic stiffness isadjusted by varying the duration of step 1. Varying the amplitude of thevoltage applied during step 1 modifies the sensitivity of detection ofthe position of the mobile electrode 5, which affects the stability ofthe feedback loop. Varying the amplitude of the voltage or the durationof step 3 also modifies the sensitivity of the accelerometer. Theelectrostatic stiffness could also be adjusted by simultaneously varyingthe amplitude of the voltages applied between the fixed electrodes 3 and4 and the mobile electrode and the duration of the position measurementstep and/or the step of application of the electrostatic force, duringtime-division multiplexing.

Adjusting the resonant frequency as described above:

compensates the spread of the mechanical stiffness of the springs 7, andthus simplifies production of the accelerometers,

reduces the sagging of vertical accelerometers, without loss ofperformance, and

optimizes performance as a function of the wanted band.

In geophysical seismic prospecting, for example, it is standard practiceto vary the bandwidth of the signal acquisition system as a function ofthe areas explored or investigation depth targets: 100, 200 or 400 Hz.If the resonant frequency of a prior art accelerometer is optimized fora given bandwidth, for example 100 Hz, the performance is very degradedif it is used at 400 Hz. Adjusting the electrostatic stiffness to thenew conditions of use, as with the accelerometer of the invention,re-establishes optimum performance.

As described above, the accelerometer according to the inventioncomprises a sensor 1 with one mobile electrode 5 and two fixedelectrodes 3 and 4 but the sensor 1 can also comprise several mobileelectrodes 5 and more than two fixed electrodes 3 or 4. Similarly, aplurality of sensors 1 can be associated with each other to constitutean accelerometer according to the invention in which the resonantfrequency of each sensor 1 is determined by a mechanical stiffness andadjustment of the electrostatic stiffness as described above.

An accelerometer according to the invention can be used for seismicprospecting, detecting impacts for safety systems such as airbags, etc.

Many devices having varied applications can include an accelerometeraccording to the present invention as described hereinafter and asclaimed hereinafter.

What is claimed is:
 1. Slaved mobile plate accelerometer usingvariations of capacitance to detect the movement of a mass, comprising:at least one pair of fixed electrodes rigidly attached to an armature,and at least one mobile electrode suspended by springs from thearmature, between the two fixed electrodes of said pair of fixedelectrodes, to form two capacitors, said mobile electrode being adaptedto move between the fixed electrodes of each pair of fixed electrodes,due to the effect of the acceleration, so causing a variation of thecapacitance of each capacitor,  characterized in that said accelerometerincludes an electronic circuit for adjusting the electrostatic stiffnessof at least one combination of a fixed electrode and a mobile electrodeand comprising a control system for detecting the variation of thecapacitance of each capacitor by applying two voltages (+V_(m),−V_(m))symmetrical about ground, and reacting by applying a feedback voltagebetween each mobile electrode and the fixed electrode facing said mobileelectrode, in that the springs have a stiffness chosen intentionally toplace the mechanical resonant frequency beyond the lop frequency of theband of interest, and in that the circuit for adjusting theelectrostatic stiffness is adapted to return the apparent resonantfrequency into the band of interest.
 2. Accelerometer according to claim1, characterized in that said accelerometer comprises two fixedelectrodes electrically insulated from each other.
 3. Accelerometeraccording to claim 1, characterized in that said accelerometer comprisesa single mobile electrode.
 4. Accelerometer according to claim 1characterized in that the electronic circuit enables time-divisionmultiplexing of each mobile electrode.
 5. Accelerometer according toclaim 2, wherein the accelerometer comprises a single mobile electrode,the electronic circuit enables time-division multiplexing of each mobileelectrode and wherein multiplexing cycle includes four steps, which are:a first step during which a voltage sample and its symmetricalcounterpart relative to ground are respectfully applied between eachfixed electrode and the mobile electrode, a second step during which thecapacitors constituted by one of said two fixed electrodes and themobile electrode and the capacitor constituted by the other fixedelectrode and the mobile electrode are discharged, a third step duringwhich a feedback voltage is applied to one or the other of thecapacitors constituted by the mobile electrode and one of the fixedelectrodes, as a function of a decision taken by the control system, anda fourth step during which the operation of the second step is repeated.6. Accelerometer according to claim 1 whose electronic circuit variesthe amplitudes of the voltages between each fixed electrode and eachmobile electrode to adjust the electrostatic stiffness.
 7. Accelerometeraccording to claim 1 whose electronic circuit adjusts the duration ofthe time-division multiplexing steps.
 8. Accelerometer according toclaim 7, characterized in that adjusting the duration of thetime-division multiplexing step adjusts the electrostatic stiffness. 9.Accelerometer according to claim 6, characterized in that adjusting theelectrostatic stiffness compensates the spread of the mechanicalstiffness of the springs.
 10. Accelerometer according to claim 6,characterized in that adjusting the electrostatic stiffness adjusts thesag when said accelerometer is in a vertical position, without loss ofperformance.
 11. Accelerometer according to claims 6, characterized inthat adjusting the electrostatic stiffness optimizes performance as afunction of the wanted band.
 12. Accelerometer according to claim 4,characterized in that the feedback voltage is applied between eachmobile electrode and the fixed electrode in the form of a pulse with afalling edge falling progressively, possibly even to zero, at the end ofthe pulse.
 13. Accelerometer according to claim 4, characterized in thatsaid accelerometer includes a parallel RC circuit for generating afeedback voltage in the form of a pulse with a steep rising edge and anexponentially decreasing falling edge.
 14. Application of anaccelerometer according to claim 1, to seismic prospecting. 15.Accelerometer according to claim 8, characterized in that adjusting theelectrostatic stiffness compensates the spread of the mechanicalstiffness of the springs.
 16. Accelerometer according to claim 8,characterized in that adjusting the electrostatic stiffness adjusts thesag when said accelerometer is in the vertical position, without loss ofperformance.
 17. Accelerometer according to claim 8, characterized inthat adjusting the electrostatic stiffness optimizes performance as afunction of the wanted band.